Theorem imaeq2 4968

Description: Equality theorem for image. (Contributed by NM, 14-Aug-1994.)

Assertion

Ref	Expression
imaeq2	|- ( A = B -> ( C " A ) = ( C " B ) )

Proof of Theorem imaeq2

Step	Hyp	Ref	Expression
1		reseq2 4904	. . 3 |- ( A = B -> ( C |` A ) = ( C |` B ) )
2	1	rneqd 4858	. 2 |- ( A = B -> ran ( C |` A ) = ran ( C |` B ) )
3		df-ima 4641	. 2 |- ( C " A ) = ran ( C |` A )
4		df-ima 4641	. 2 |- ( C " B ) = ran ( C |` B )
5	2, 3, 4	3eqtr4g 2235	1 |- ( A = B -> ( C " A ) = ( C " B ) )

Syntax hints:   -> wi 4   = wceq 1353   ran crn 4629   |` cres 4630   "cima 4631

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2741  df-un 3135  df-in 3137  df-ss 3144  df-sn 3600  df-pr 3601  df-op 3603  df-br 4006  df-opab 4067  df-xp 4634  df-cnv 4636  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641

This theorem is referenced by:  imaeq2i  4970  imaeq2d  4972  ssimaex  5579  ssimaexg  5580  isoselem  5823  f1opw2  6079  fopwdom  6838  ssenen  6853  fiintim  6930  fidcenumlemrk  6955  fidcenumlemr  6956  sbthlem2  6959  isbth  6968  ennnfonelemp1  12409  ennnfonelemnn0  12425  ctinfomlemom  12430  ctinfom  12431  tgcn  13747  iscnp4  13757  cnpnei  13758  cnima  13759  cnconst2  13772  cnrest2  13775  cnptoprest  13778  txcnp  13810  txcnmpt  13812  metcnp3  14050